The stable rank of $\mathbb{Z}[x]$ is $3$

TL;DR

This paper corrects minor errors in the proof that the Bass stable rank of the polynomial ring over integers is 3, and demonstrates a specific unimodular row that is not stable.

Contribution

It refines the proof of the stable rank of 2[x] and provides a concrete example of a non-stable unimodular row.

Findings

The Bass stable rank of 2[x] is 3.

A specific unimodular row 2, x+1, x^2+16 is not stable.

Minor errors in previous proof are addressed.

Abstract

Grunewald, Mennicke and Vaserstein proved that the Bass stable rank of $\mathbb{Z}[x]$, the ring of the univariate polynomials over $\mathbb{Z}$, is $3$. This note addresses minor errors found in their proof. Using their method, we show in addition that the unimodular row $(3, x + 1, x^2 + 16)$ is not stable.